\magnification = 2000 
\input amstex
\documentstyle{amsppt}
\input OurATOMacros
\input OurPlainGraphicsMacros

\vglue -30pt

\Title{Bianchi-Pinkall Flat Tori in $\Bbb S^3$}.
\section{Parameter Dependent Formulas in 3DXM}
\vskip0.3mm\noindent
 We can parametrize $\Bbb S^3$, considered as a submanifold of $\Bbb C^2$, by:
$$ F(u,\alpha,v) = (\cos(\alpha) e^{iu} e^{iv} , \sin(\alpha) e^{iu} e^{-iv}),$$
 where  $u \in [0,2 \pi)$, $\alpha \in [0, \pi/2]$, and $v \in [0, \pi]$.  
We will get the Pinkall Tori first as flat tori in $\Bbb S^3$ by taking $\alpha$ to be a 
 function of $v$, $\alpha := aa + bb \sin(\hbox{ee}\, 2v)$ (although the theory allows 
 more general choices.)  Next we stereographically project $\Bbb S^3$ from \lf
\cl   {$p = (\cos(cc\cdot\pi),0,\sin(cc \cdot\pi),0)$} 

\noindent
 to get {\bf conformal} images of the flat  tori in $\Bbb S^3$. The lines $v = const$ are
 circles, the stereographic images of the Hopf circles $u\mapsto F(u,\alpha,v)$.
 \smallskip
  \cl {\includegraphics[width=2.0in]{BianchiPinkal3.png}}
 \noindent %\lf
Finally, by morphing $0 \le f\!f \le 2\pi$, we can isome\-trically rotate  $\Bbb S^3$ so that
the Hopf circle $v=0$ is the rotation axis. The stereographic image of this rotation
is a conformal transformation of $\Bbb R^3 \cup \{\infty\}$ which ``rotates'' $\Bbb R^3$
around a circle on the pictured torus. In the case $aa =\pi/4$ we obtain for $f\!f=0$ and
$f\!f=\pi$ the same torus, but inside and outside interchanged. This is best viewed
with the default `Two Sided User Coloration'. It can be selected from a submenu of the
Action Menu.

\smallskip
\section{Background and Explanations}
\vskip0.3mm\noindent
The tori that we usually see are, from the point of view
of complex analysis, rectangular tori,  meaning that they have
an orientation reversing symmetry and the set of fixed points of this
symmetry has two components. (The better known tori of revolution have
isometric reflections with {\bf two} circles as fixed point sets.)  Of
course one tries to deform these tori to obtain non-rectangular ones.
Obviously one can destroy the mirror symmetry, but this does not imply
that one gets tori with a non-rectangular complex structure. The first
proof, by Garcia, that one can embed all tori in $\Bbb R^3$ was
non-constructive and difficult. 
\Lf
A simpler and constructive way to get tori with arbitrary conformal type was found by Pinkall,
whose idea was to construct tori that are flat in $\Bbb S^3$ (and hence have an easy way 
to compute their conformal type from their flat geometry), and then stereographically 
project them to $\Bbb R^3$.  While the resulting tori are no longer flat, this does preserve their 
conformal type.
\Lf
The construction of flat surfaces in $\Bbb S^3$ goes back to 1894, when Bianchi classified all flat immersions in $\Bbb S^3$.
In particular, he realized that the two families  of asymptotic lines of a flat surface in $\Bbb S^3$ are left translations of a pair of 
curves that are either great circles or have constant torsion $+1$ and $-1$, respectively. 
The left translations arise by viewing $\Bbb S^3$ as the group of unit quaternions.
An open problem for Bianchi was to determine when his flat surfaces were closed.
\Lf
The first case when one of the curves is a great circle is of special interest for this problem. To explain why, we will need the
Hopf fibration. Thinking of $\Bbb S^3$ as being part of $\Bbb C^2$, we can multiply points of $\Bbb S^3$ by $e^{i u}$, thus fibering  
$\Bbb S^3$ with circles, the Hopf circles, and the set of all such circles forms a metric space with distance being the distance between the Hopf 
circles in $\Bbb S^3$. As such it is isometric to a 2-dimensional sphere of radius $1/2$. We thus obtain a 
natural projection $\Bbb S^3\to\Bbb S^2$, the Hopf map.
It can be written as $(z_1,z_2)\mapsto z_1/z_2$, where we interpret the range as the Riemann sphere $\hat {\Bbb C}$. 
Moreover,  Hopf circles are mapped to Hopf circles by left translations.
\Lf
Now suppose we have a flat surface in $\Bbb S^3$ where one of the generating curves is a great circle. We can arrange $\Bbb S^3$ so that 
this great circle is part of the Hopf fibration, and thus all curves of the same family of asymptotic lines are Hopf circles.
 The surface in $\Bbb S^3$ is thus invariant under the Hopf action and projects to a curve in $\Bbb S^2$ under the Hopf map. Vice versa, 
 the preimage of a curve in $\Bbb S^2$ under the Hopf map
yields a flat surface in $\Bbb S^3$. In case the curve in $\Bbb S^2$ is closed, the surface in $\Bbb S^3$ is a flat torus. 
(The explanation so far is described in more detail in Spivak IV, p. 139ff.)
\Lf
Pinkall found a simple way to determine the conformal type of the flat torus in terms of the geometry of the curve in 
$\Bbb S^2$ --- in particular it was then easy to see that {\bf all} possible conformal types can occur.
\LF
\section{Visualizing Parts of the Theoretic Description}
\vskip0.3mm\noindent
We cannot visualize $\Bbb S^3$ in such a way
 that all distances are preserved. We will use stereographic
projection from $p = (\cos(cc\cdot\pi),0,\sin(cc \cdot\pi),0)$ to map $\Bbb S^3 - \{p\}$ one-to-one
onto $\Bbb R^3$. Recall that: angles are not changed by stereographic projection, circles are
mapped to circles or straight lines, and the images of great circles meet the equator sphere in
antipodal points, so many properties of $\Bbb S^3$ get represented in geometrically comprehensible
ways.
 \lf
 Our parametrization $F$ of $\Bbb S^3$ emphasizes the Hopf fibration since
 the great circles $u \mapsto F(u,\alpha,v)$ are indeed the orbits of the Hopf-action
 of $\Bbb S^1$ on $\Bbb S^3$, given by $(u,p)\mapsto e^{iu} p$. Each such ``Hopf Fiber'' 
lies in one of the parallel tori  $\alpha = $ constant, and the great circles 
$\alpha \mapsto F(u,\alpha,v)$, meet these ($\alpha = $ constant)-tori orthogonally, 
so that $\alpha$ measures the distance between them. 
\lf
We get all the Hopf circles on each $\alpha$-torus for $0 \le v \le \pi$, except that those
tori degenerate to just one Hopf circle if $\alpha = 0$ or $\alpha = \pi$. This makes it plausible
that  $(\alpha,2v)$ are polar coordinates on the metric space of Hopf circles, on the image
 $\Bbb S^2$ of the Hopf map.
\lf
Pinkall has observed that the closed curves on this image sphere, in polar coordinates given as:
$(\alpha(s),2v(s))$,  (with $\alpha(s)$ never equal to $0$ or $\pi/2$) allow one to write down
immersed tori  in $\Bbb S^3$ as:
\lf 
\cl{$(u,s) \mapsto (F(u,\alpha(s),v(s)).$}

\noindent
For example taking $\alpha(s) = \pi/4$ gives the ``Clifford Torus'' in $\Bbb S^3$, a minimal 
embedding of the square torus. For other constant $\alpha(s)$ in $(0,\pi/2)$ one gets the
above parallel family of $\alpha$-tori, the lengths of their
two orthogonal generators are $2 \pi \cos(\alpha) $ and $2 \pi \sin(\alpha)$.
 
\noindent
On all of these tori we still have that the parameter lines $s=$ constant are Hopf-Fibers, 
and since these are equi\-distant (as orbits of an isometric action) it follows that
the metric is flat. Pinkall proved that  length and area of the curve in 
$\Bbb S^2$ determine the con\-formal structure of the torus in $\Bbb S^3$, hence in $\Bbb R^3$,
and that all conformal structures occur.
 
\noindent
Observe that the usual tori of revolution in $\Bbb R^3$ are all rectangular, and
most of the Pinkall tori shown by 3D-XplorMath are very different from these.
The tori with aa $= \pi/4$ are all rhombic, because they can be rotated into themselves 
by $180^\circ$ rotations (in $\Bbb S^3$, not in $\Bbb R^3$) around any of the Hopf-Fibers
on them. A cyclic morph with $ 0 \le f\!f  \le 2\pi$ rotates around the circle $v=0$
(we see of course the stereographic image of that rotation). For $f\!f=\pi$ we get
an anti-involution of the torus with the circle as the (connected) fixed point set---only 
rhombic tori have such anti-involutions. (The square torus is rectangular and rhombic.)
In the rhombic case $aa=\pi/4$ we get
for $f\!f=\pi/2$ and $f\!f=3\pi/2$ surfaces in $\Bbb S^3$ that pass through $p$ so
that the stereographic images in $\Bbb R^3$ pass through $\infty$ --- otherwise we could
not turn the torus inside out continuously.
 
\noindent
The program takes $\alpha(v) := aa + bb \sin(\hbox{ee}\, 2v)$ (with $ee= 3$ for the default
image and $ee= 5$ for the default morph), allowing rather different examples.
\lf Again, these tori are
shown in $\Bbb R^3$ by using the (conformal) stereographic projection of 
$\Bbb S^3\setminus\{p\} \to \Bbb R^3$, where  {$p = (\cos(cc\cdot\pi),0,\sin(cc \cdot\pi),0)$} .
Morphing $cc$ therefore gives other images of $\Bbb S^3$, in particular other
conformal images of these tori.

\noindent
H.K.
 




\bye
